Lunar and Planetary Science XLVIII (2017)
1017.pdf
A MODEL OF HAZARDOUS ASTEROIDS MOTION. N. I. Perov1 , 1Cultural and Educational Center named
after V.V. Tereshkova, ul. Chaikovskogo, 3, Yaroslavl, 150000, Russian Federation. E-mail: [email protected].
Introduction: In the work [1] the model of
hazardous comets migration from Oort’s and Hill’s
clouds has been considered in the frame of planar
circular three body problem “the Sun - small body with
mass about 10-100 times less then mass of the Moon a cometary nucleus with negligible mass”. Below in
the frame of five body problem “the Sun - the Earth the Moon - Mars - body with negligible mass” motion of a hazardous asteroid (and which is a marscrosser) from main belt of asteroids is investigated.
Based Equation: Let’s consider in the frame of a
planar five body problem motion of a hazardous
asteroid in the gravitational fields of the Sun, the Earth,
the Moon, and Mars. In some cases it is convenient to
represent the vector differential equation of the
hazardous asteroids motion in the form (1) with one
independent variable vE [3].
(d2r/dvE2)ωE2 = -GmSr/r3-GmE (r-rE) /│r-rE│3- GmM
(r-rE-rEM) /│r-rE-rEM │3-GmA (r-rA) /│r-rA│3.
(1)
Here, r, rE, rA are heliocentrically vectors of the
asteroid, Mars and
the Earth positions,
correspondingly. rEM is the vector of the Moon
geocentrically position. mS, mE, mM, and mA are mass
of the Sun, the Earth, the Moon and Mars,
consequently. vE is a true anomaly of the Earth, ωE is
the angular velocity of the Earth that uniformly rotates
around the Sun along the circle orbit. The orbits of the
Moon and Mars are circles and theirs true anomalies
are proportional to vE. In this celestial mechanical
model the true anomalies vE, vM, vA of the Earth, Moon
and Mars, correspondingly, are equal to theirs mean
anomalies of these bodies. G is the gravitational
constant. Radii of the Earth, the Moon and Mars are
equal to RE=0.000042638 AU, RM=0.00001162 AU
and RA =0.00002271 AU correspondingly.
Examples: Put for the ratios of the orbital periods of
the Earth and the Moon; the Earth and the Mars the
following values: k=13.368878522 and m =
0.53171491747, then
vM=vEk+vM0 ,
(2)
vA=vEm+vA0.
(3)
The origins of the true anomalies readings are
coincided with initial positions of the bodies. We use
the following system of units. The unit of length is one
astronomical unit. The unit of mass is mass of the Sun.
The unit of time is 1 sidereal year. vE is measured in
radians. For the initial conditions equal vE0=0.949643
rad, vM0=2.021 rad, vA0=4.1411821 rad, x0=3 AU,
y0=0, Vx0=0, Vy0=11000 m/s the trajectory of the
hazardous asteroid is presented in Fig.1. The variations
of the distance between the Sun and the asteroid are
shown in the Fig.2. The variations of the distance
between the Earth and the asteroid are shown in the
Fig.3. and Fig.4. The variations of the distance
between Mars and the asteroid are shown in the Fig.5.
The variations of the distance between the Moon and
the asteroid are presented in the Fig.6. Figure 4 and
figure 6 are insignificantly distinguished.
Fig.1. The heliocentrically trajectory of the
hazardous asteroid passed near Mars, the Moon and the
Earth. vEmax=60 rad.
Fig.2. The distance between the Sun and the
hazardous asteroid in astronomical units. vEmax=60 rad.
Lunar and Planetary Science XLVIII (2017)
Fig.3. The distance rEH between the Earth and the
hazardous asteroid. The minimal distance equals
0.00044591 AU (>RE) for vE=7.4107855 rad.
VEmax=2000 rad. For vE=255.8517 rad we found
rEH=0.0047106 AU and for vE=538.575 rad we found
rEH=0.026135 AU. (rEH>RE)
Fig.4. The distance between the Earth and the
hazardous asteroid. The minimal distance equals
0.00044591 AU (>RE) for vE=7.4107855 rad. vEmax=60
rad.
1017.pdf
Fig.5. The distance between Mars and the
hazardous asteroid (mars-crosser). The minimal
distance equals 0.0000228945 AU (>RA) for
vE=6.34007417 rad. vEmax=60 rad.
Fig.6. The distance between the Moon and the
hazardous asteroid. The minimal distance equals
0.002327 AU (>>RM) for vE=7.4082926 rad. vEmax=60
rad.
Conclusions: 1. the equation (1) is the equation
with one unknown function depended from one
independent variable vE. 2. Some asteroids appear near
the Earth like hazardous bodies after passing near the
surface of Mars, but corresponding “keyholes” [2] are
small. 3. The “keyholes” [2] are especially too small
for the asteroids passing near the Moon. 4. The
considered model may be useful for the monitoring of
the surfaces and studying the Moon and Mars with
help by one cosmic apparat.
References: [1] Perov N.I. (2016) LPS XLVII,
Abstract #1010. [2] Chodas P.W. (2012) ACM.
Abstract #6471. [3] Roy A. (1978) Orbital motion
Adam Higler Ltd. Bristol. 545 pp.